Thermodynamics extensive variables

Before describing these thermodynamic variables, we must talk about their properties. The variables are classified as intensive or extensive. Extensive variables depend upon the amount while intensive variables do not. Density is an example of an intensive variable. The density of an ice crystal in an iceberg is the same as the density of the entire iceberg. Volume, on the other hand, is an extensive variable. The volume of the ocean is very different from the volume of a drop of sea water. When we talk about an extensive thermodynamic variable Z we must be careful to specify the amount. This is usually done in terms of the molar property Zm, defined as... 

In addition to the fundamental variables p, V, T, U, and S that we have described so far, three other thermodynamic variables are commonly encountered enthalpy Helmholtz free energy and Gibbs free energy. They are extensive variables that do not represent fundamental properties of the... 

The substitutions can be made because the extensive thermodynamic variables in the equations are homogeneous of degree one.d Thus, dividing the equation by n converts the extensive variable to the corresponding molar intensive variable. For example, to prove that equation (3.48) follows from equation... 

The relationships summarized in Table 3.1. expanded to include differences and molar properties, serve as the starting point for many useful thermodynamic calculations. An example is the calculation of AZ for a variety of processes in which p, V, and T are changed.e For any of the extensive variables Z = S, U, H, A or G, we can write... 

First, we note that all of the thermodynamic equations that we have derived for the total extensive variables apply to the partial molar properties. Thus, if... 

The extensive variable Q associated with the electrical potential + in Eqs. (15), (17), and (21) is the thermodynamic surface excess charge density, which is defined by... 

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

In thermodynamics the state of a system is specified in terms of macroscopic state variables such as volume, V, temperature, T, pressure,/ , and the number of moles of the chemical constituents i, tij. The laws of thermodynamics are founded on the concepts of internal energy (U), and entropy (S), which are functions of the state variables. Thermodynamic variables are categorized as intensive or extensive. Variables that are proportional to the size of the system (e.g. volume and internal energy) are called extensive variables, whereas variables that specify a property that is independent of the size of the system (e.g. temperature and pressure) are called intensive variables. 

In general, dw is written in the form (intensive variable)-d(extensive variable) or as a product of a force times a displacement of some kind. Several types of work terms may be involved in a single thermodynamic system, and electrical, mechanical, magnetic and gravitational fields are of special importance in certain applications of materials. A number of types of work that may be involved in a thermodynamic system are summed up in Table 1.1. The last column gives the form of work in the equation for the internal energy. 

The analogue to one-component thermodynamics applies to the nature of the variables. So Ay S, U and V are all extensive variables, i.e. they depend on the size of the system. The intensive variables are n and T -these are local properties independent of the mass of the material. The relationship between the osmotic pressure and the rate of change of Helmholtz free energy with volume is an important one. The volume of the system, while a useful quantity, is not the usual manner in which colloidal systems are handled. The concentration or volume fraction is usually used ... 

In the study of thermodynamics we can distinguish between variables that are independent of the quantity of matter in a system, the intensive variables, and variables that depend on the quantity of matter. Of the latter group, those variables whose values are directly proportional to the quantity of matter are of particular interest and are simple to deal with mathematically. They are called extensive variables. Volume and heat capacity are typical examples of extensive variables, whereas temperature, pressure, viscosity, concentration, and molar heat capacity are examples of intensive variables. 

The advantage of the chemical potential over the other thermodynamic quantities, U, H, and G, is that it is an intensive quantity—that is, is independent of the number of moles or quantity of species present. Internal energy, enthalpy, free energy, and entropy are all extensive variables. Their values depend on the extent of the system—that is, how much there is. We will see in the next section that intensive variables such as p., T, and P are useful in defining equilibrium. 

Although gibbsite and kaolinite are important in quantity in some soils and hydrothermal deposits, they have diminishing importance in argillaceous sediments and sedimentary rocks because of their peripheral chemical position. They form the limits of any chemical framework of a clay mineral assemblage and thus rarely become functionally involved in critical clay mineral reactions. This is especially true of systems where most chemical components are inert or extensive variables of the system. More important or characteristic relations will be observed in minerals with more chemical variability which respond readily to minor changes in the thermodynamic parameters of the system in which they are found. However, as the number of chemical components which are intensive variables (perfectly mobile components) increases the aluminous phases become more important because alumina is poorly soluble in aqueous solution, and becomes the inert component and the only extensive variable. 

Consider next the energy equation, neglecting kinetic and gravitational-potential energy. Here the extensive variable is the internal energy of the gas E and the intensive variable is the specific internal energy e. The first law of thermodynamics provides the system energy balance... 

It should be emphasized that the criterion for macroscopic character is based on independent properties only. (The importance of properly enumerating the number of independent intensive properties will become apparent in the discussion of the Gibbs phase rule, Section 5.1). For example, from two independent extensive variables such as mass m and volume V, one can obviously form the ratio m/V (density p), which is neither extensive nor intensive, nor independent of m and V. (That density cannot fulfill the uniform value throughout criterion for intensive character will be apparent from consideration of any 2-phase system, where p certainly varies from one phase region to another.) Of course, for many thermodynamic purposes, we are free to choose a different set of independent properties (perhaps including, for example, p or other ratio-type properties), rather than the base set of intensive and extensive properties that are used to assess macroscopic character. But considerable conceptual and formal simplifications result from choosing properties of pure intensive (R() or extensive QQ character as independent arguments of thermodynamic state functions, and it is important to realize that this pure choice is always possible if (and only if) the system is macroscopic. 

Each such null vector may be considered an invariant or symmetry of the thermodynamic system, because it corresponds to an operation (change of extensive variables Xt) that produces no response in any intensive state variable and thus leaves the thermodynamic state unaltered (Sidebar 7.2). As described in Sidebar 10.3, these invariants also play a role somewhat analogous to overall rotations and translations ( null eigenmodes of the Hessian matrix) in the theory of molecular vibrations. 

It is always convenient to use intensive thermodynamic variables for the formulation of changes in energetic state functions such as the Gibbs energy G. Since G is a first order homogeneous function in the extensive variables V, S, and rtk, it follows that [H. Schmalzried, A.D. Pelton (1973)]... 

Here V(m ) is the probability distribution for the generalized mean size in the first phase, taken over partitions with fixed and N with equal a priori probabilities. Note that given m, irP is fixed in the second phase by the moment equivalent of particle conservation iV W1) + N mPl = Nm(° The integral in (17) can be replaced by the maximum of the integrand in the thermodynamic limit, because In V(m ) is an extensive quantity. Introducing a Lagrange multiplier pm for the above moment constraint then shows that the quantity pm has the same status as the density p = p0 itself Both are thermodynamic density variables. This reinforces the discussion in the introduction, where we showed that moment densities can be regarded as densities of quasi-species of particles. 

Finally, the thermodynamic properties of a system considered as variables may be classified as either intensive or extensive variables. The distinction between these two types of variables is best understood in terms of an operation. We consider a system in some fixed state and divide this system into two or more parts without changing any other properties of the system. Those variables whose value remains the same in this operation are called intensive variables. Such variables are the temperature, pressure, concentration variables, and specific and molar quantities. Those variables whose values are changed because of the operation are known as extensive variables. Such variables are the volume and the amount of substance (number of moles) of the components forming the system. 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

The beauty of the fundamental equation for U (equation 2.2-8) is that it combines all of this information in one equation. Note that the Ns + 2 extensive variables S, V, and n, are independent, and the Ns + 2 intensive variables T, P, and /uj obtained by taking partial derivatives of U are dependent. This is wonderful, but equations of state 2.2-10 to 2.2-12 are not very useful because S is not a convenient independent variable. Fortunately, more useful equations of state will be obtained from other thermodynamic potentials introduced in Section 2.5. 

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

The internal energy is homogeneous of degree 1 in terms of extensive thermodynamic properties, and so equation 2.2-8 leads to equation 2.2-14. All extensive variables are homogeneous functions of the first degree of other extensive properties. All intensive properties are homogeneous functions of the zeroth degree of the extensive properties. 

Chemical thermodynamics deals with the physicochemical state of substances. All physical quantities corresponding to the macroscopic property of a physicochemical system of substances, such as temperature, volume, and pressure, are thermodynamic variables of the state and are classified into intensive and extensive variables. Once a certain number of the thermodynamic variables have been specified, then all the properties of the system are fixed. This chapter introduces and discusses the characteristics of intensive and extensive variables to describe the physicochemical state of the system. 

An extensive variable may be converted into an intensive variable by expressing it per one mole of a substance, namely, by partially differentiating it with respect to the number of moles of a substance in the system. This partial differential is called in chemical thermodynamics the partial molar quantity. For instance, the volume vi for one mole of a substance i in a homogeneous mixture is given by the derivative (partial differential) of the total volume V with respect to the number of moles of substance i as shown in Eq. 1.3 ... 

In these equations we see the regularity that the partial differential of these four thermodynamic potentials with respect to their respective extensive variables gives us their conjugated intensive variables and vice versa. We thus obtain minus the affinity of an irreversible process in terms of the partial differentials of U, H, F, and G with respect to the extent of reaction affinity is an extensive variable. 

The values of extensive thermodynamic variables, such as N, V, and E, are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size of the system. Show that entropy is an extensive variable. 

The 1 IT and fi/T appear in the thermodynamic conjugates of the extensive variables in the Gibbs equation for the system entropy... 

Intensive variable. A measurable property of a thermodynamic system is intensive if when two identical systems are combined into one, the variable of the combined system is the same as the original value in each system. Examples temperature, pressure. See extensive variable, and specific. 

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